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As background, I briefly review the progression of steps that culminated in the Extended Mind Hypothesis, and allude to the controversies it raised.

Chapter Background | Nonlinear Science | The National Academies Press

Assuming the validity of this hypothesis, I will explore the issues that arise from viewing brain, body and world as ONE complex dynamical system. This will lead me to suggest that interrelations between complex system and fractal dynamics enable the seamless integration of human capabilities and the material world.

To perceive, and more particularly to sense, is conditional upon the state of uncertainty of the perceiving system. This idea is developed from its philosophical base, to the equations that encode it, and through to the experimental data that provide its support. Much of the paper constitutes a review of research conducted over past years, but viewed now with some perspective.

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This uncertainty concept represents a kind of thermodynamic principle, which tells us how various sensory phenomena will occur, but does not provide a physiological mechanism. Hence, for example, the mathematical structure of the Weber-Fechner law will emerge, but it remains for mechanisms such as, perhaps, neural networks, to define exactly how the law will be implemented physiologically.

The latter part of the paper deals with the attempt to link sensation, for one special case, to the physical entropy of the received signal. The unique dynamical features of the critical state can endow the brain with properties which are fundamental for adaptive behavior.


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This proposal, put forward with Per Bak several years ago, is now supported by a wide body of empirical evidence at different scales demonstrating that the spatiotemporal brain dynamics exhibits key signatures of critical dynamics previously recognized in other complex systems. The rationale behind this program is discussed in these notes, followed by an account of the most recent results, together with a discussion of the physiological significance of these ideas.

The Bak—Sneppen model demonstrates that extremal selection of poorly adapted variables, which are interconnected in a system, can lead to a highly correlated state that is both, well-adapted and responsive. We have developed a solvable version of the model that provides detailed insights in how memory is acquired in the system. We have employed the Bak—Sneppen dynamics to design a highly efficient optimization heuristic, called "Extremal Optimization" EO , to find low-energy or low-cost configurations for many combinatorial problems in physics and computer science. Analysis of the local search dynamics induced by EO reveals that the configuration space is traversed most efficiently at the "edge of ergodicity" which effectively exploits the long-range memory encapsulated in the strongly correlated state of the variables.

There, the dynamics of local search possesses a generic critical point under the variation of its sole parameter, separating phases of too greedy non-ergodic, jammed and too random ergodic, diffusive exploration. Analytic comparison with other local search methods, such as a fixed temperature Metropolis algorithm, within this model suggests that the existence of the critical point is the essential distinction leading to the optimal performance of the extremal optimization heuristic.

We study random walks on networks, either regular or complex, and we show that the observation of a single walker is not quite adequate to properly determine the network efficiency, which is expected to increase with the network topological complexity.


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  7. In fact, the time complexity of a single walker is compatible with the two opposite conditions, of regular topology, with minimum efficiency, and of scale-free topology, with rich nodes that are linked to all the other nodes of the network, thereby generating the condition of maximum efficiency. The observation of the motion of a large number of walkers leads to unambiguous results, with the former limiting condition yielding equilibrium with a time scale proportional to the number of nodes, N , and the latter limiting condition corresponding to an almost instantaneous transition to equilibrium, more precisely with a time scale proportional to log N.

    We study a model of cooperating units, each of which is a stochastic clock, and we show that when the cooperation strength is critical the model yields coherence and complexity at the same time. We conjecture concerning the relevance of this model to understanding brain dynamics. In this Chapter we study a set of cooperating neurons generating quakes with the same size distribution as the neuronal avalanches in mature cultured networks, recently revealed by the experimental observation.

    We prove that in addition to this form of complexity this model yields a form of phase transition generating also temporal complexity. This means that the distance from two consecutive quakes fits the prescription of a Mittag-Leffler ML function renewal theory.

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    A common theme among these and many other related lectures is to model, study, understand, and exploit the rich behavior exhibited by nonlinear systems to design and fabricate novel technologies with superior characteristics. In spite of significant advances in material properties, in many cases it remains a daunting task to duplicate the superior signal processing capabilities of most animals.

    Since nonlinear systems tend to be highly sensitive to perturbations when they occur near the onset of a bifurcation, there are also lectures on the general topic of bifurcation theory and on how to exploit such bifurcations for signal enhancements purposes. This manuscript will appeal to researchers interested in both theory and implementations of nonlinear systems. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser.

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